2
$\begingroup$

The topological susceptibility in QCD (here I've used path integral approach, and hence I will neglect all contact terms) is defined as $$ \kappa (p) \equiv \lim_{y \to 0}\int d^{4}x e^{ip(x-y)}Q(x)Q(y)e^{iS} \equiv \lim_{y \to 0}\int d^{4}xe^{ip(x-y)}\langle 0| Q(x)Q(y)|0\rangle , $$ where $$ Q(x) \equiv \frac{1}{32 \pi}G \tilde{G} \equiv \partial_{\mu}^{x}K^{\mu}(x) $$ is the topological charge of QCD.

It can be shown, after some manipulations, that $$ \tag 1 \kappa (0) = -c\langle 0|\bar{u}u|0\rangle , $$ where $c$ is constant (which is zero if at least one of quarks is massless).

From the other hand, from the definition we have that $$ \tag 2 \kappa_{0} \equiv \lim_{p \to 0}p_{\alpha}p_{\beta}\Pi^{\alpha \beta}(p), $$ where $$ \Pi^{\alpha \beta}(p) \equiv \int d^{4}x e^{ipx} \langle 0| K^{\alpha}(x)K^{\beta}(0)|0\rangle $$ By combining $(1)$ and $(2)$ we have the statement that $$ \lim_{p \to 0}p_{\alpha}p_{\beta}\Pi^{\alpha \beta}(p) = -c\langle 0| \bar{u}u|0\rangle \neq 0 $$ So we conclude that $P^{\alpha \beta}(p)$ has the pole. However, this doesn't fixed uniquely the pole structure of $P^{\alpha \beta}$: in general, $$ \lim_{p \to 0}\Pi_{\alpha \beta}(p) = -a\frac{g_{\alpha \beta}}{p^{2}} + b\frac{p_{\alpha}p_{\beta}}{p^{4}}, \quad a + b = c\langle 0| \bar{u}u|0\rangle $$ This ambiquity is important, since if $a \neq 0$, then, for example, the gluon propagator obtains positive valued correction in denominator, $$ \tag 3 \lim_{p \to 0} D_{\mu \nu}(p) \sim \frac{g_{\mu \nu}}{p^{2} + \frac{a\kappa (0)}{p^{2}}}, $$ which describes confinement phenomena qualitatively, while if $a = 0$, then there is no correction.

So how to establish the value of $a$ in $(3)$?

$\endgroup$
  • $\begingroup$ Ia the physical meaning of topological correlator $\Pi_{\alpha\beta}$ something to do with the instanton tunneling event? [Because the $Q$ is related to the instanton number, and $K^\alpha$ looks like its current?] $\endgroup$ – annie heart May 29 '18 at 1:55
3
$\begingroup$

1) Note that $p^\alpha p^\beta \Pi_{\alpha\beta}$ is gauge invariant, but $\Pi_{\alpha\beta}$ is not. This implies that $a$ and $b$ are not separately gauge invariant, only their sum (or difference, in your convention) is. For example, Diakonov and Eides [Sov. Phys. JETP 54 (2), 232-240] write down a spectral representation corresponding to $a=0$, but many other authors implicitly assume $b=0$.

2) Also note that there is no model independent relation between the gluon propagator and the topological correlator $\Pi_{\alpha\beta}$.

$\endgroup$
  • $\begingroup$ Ia the physical meaning of topological correlator $\Pi_{\alpha\beta}$ something to do with the instanton tunneling event? [Because the $Q$ is related to the instanton number, and $K^\alpha$ looks like its current?] $\endgroup$ – annie heart May 29 '18 at 1:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.